Topological gauge theories and group cohomology


We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH 4(BG,Z). In a similar way, possible Wess-Zumino interactions of such a groupG are classified byH 3(G,Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models involves a certain natural map fromH 4(BG,Z) toH 3(G,Z). We generalize this correspondence to topological “spin” theories, which are defined on three manifolds with spin structure, and are related to what might be calledZ 2 graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.

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Communicated by A. Jaffe

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Dijkgraaf, R., Witten, E. Topological gauge theories and group cohomology. Commun.Math. Phys. 129, 393–429 (1990).

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